Recursive functionals and quantifiers of finite types revisited. V

Author:
S. C. Kleene

Journal:
Trans. Amer. Math. Soc. **325** (1991), 593-630

MSC:
Primary 03D65

DOI:
https://doi.org/10.1090/S0002-9947-1991-0974519-8

MathSciNet review:
974519

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Abstract: This is the last in a sequence of papers that redoes the theory of recursion in finite types. A key feature of the theory is that a computation can succeed (or finish) even if some of its subcomputations do not, if these turn out to be irrelevant to the total computation. I give a detailed description of computations involving oracles for type functionals. The computation may be viewed formally as a transfinite sequence of symbolic expressions, but I also describe a semantics in which each expression is given a concrete realization.

**[1978**S. C. Kleene,*Recursive functionals and quantifiers of finite types revisited*, I]*Recursive functionals and quantifiers of finite types revisited. I*, Generalized recursion theory, II (Proc. Second Sympos., Univ. Oslo, Oslo, 1977) Stud. Logic Foundations Math., vol. 94, North-Holland, Amsterdam-New York, 1978, pp. 185–222. MR**516936****[1980**S. C. Kleene,*Recursive functionals and quantifiers of finite types revisited*, II, The Kleene Symposium (Madison, Wis]*Recursive functionals and quantifiers of finite types revisited. II*, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Stud. Logic Foundations Math., vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 1–29. MR**591873****[1982**S. C. Kleene,*Recursive functionals and quantifiers of finite types revisited*, III, Patras Logic Symposion (Patras, Greece, August 18-22, 1980) (G]*Recursive functionals and quantifiers of finite types revisited. III*, Patras Logic Symposion (Patras, 1980) Stud. Logic Foundations Math., vol. 109, North-Holland, Amsterdam-New York, 1982, pp. 1–40. MR**694251****[1985**Sympos. Pure Math, (from the American Mathematical Society's 1982 Summer Research Institute on Recursion Theory at Cornell University, June 27-July 16) (A. Nerode, ed.), vol. 42, Amer. Math. Soc., Providence, R.I., pp. 119-138.*Unimonotone functions of finite types*(*Recursive functionals and quantifiers of finite types revisited*, IV), Proc]

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DOI:
https://doi.org/10.1090/S0002-9947-1991-0974519-8

Article copyright:
© Copyright 1991
American Mathematical Society